Trigonometry Examples

$4 \sin^{2} (x) = 9 \cos^{2} (x) + 6 \cos (x) + 1$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $9 \cos^{2} (x)$ from both sides of the equation.

$4 \sin^{2} (x) - 9 \cos^{2} (x) = 6 \cos (x) + 1$

Step 1.2

Subtract $6 \cos (x)$ from both sides of the equation.

$4 \sin^{2} (x) - 9 \cos^{2} (x) - 6 \cos (x) = 1$

Step 1.3

Subtract $1$ from both sides of the equation.

$4 \sin^{2} (x) - 9 \cos^{2} (x) - 6 \cos (x) - 1 = 0$

Step 2

Replace the $4 \sin^{2} (x)$ with $4 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$4 (1 - \cos^{2} (x)) - 9 \cos^{2} (x) - 6 \cos (x) - 1 = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$4 \cdot 1 + 4 (- \cos^{2} (x)) - 9 \cos^{2} (x) - 6 \cos (x) - 1 = 0$

Step 3.2

Multiply $4$ by $1$ .

$4 + 4 (- \cos^{2} (x)) - 9 \cos^{2} (x) - 6 \cos (x) - 1 = 0$

Step 3.3

Multiply $- 1$ by $4$ .

$4 - 4 \cos^{2} (x) - 9 \cos^{2} (x) - 6 \cos (x) - 1 = 0$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $1$ from $4$ .

$- 4 \cos^{2} (x) - 9 \cos^{2} (x) - 6 \cos (x) + 3 = 0$

Step 4.2

Subtract $9 \cos^{2} (x)$ from $- 4 \cos^{2} (x)$ .

$- 13 \cos^{2} (x) - 6 \cos (x) + 3 = 0$

Step 5

Substitute $u$ for $\cos (x)$ .

$- 13 {(u)}^{2} - 6 u + 3 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = - 13$ , $b = - 6$ , and $c = 3$ into the quadratic formula and solve for $u$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 13 \cdot 3)}}{2 \cdot - 13}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 6$ to the power of $2$ .

$u = \frac{6 \pm \sqrt{36 - 4 \cdot - 13 \cdot 3}}{2 \cdot - 13}$

Step 8.1.2

Multiply $- 4 \cdot - 13 \cdot 3$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 13$ .

$u = \frac{6 \pm \sqrt{36 + 52 \cdot 3}}{2 \cdot - 13}$

Step 8.1.2.2

Multiply $52$ by $3$ .

$u = \frac{6 \pm \sqrt{36 + 156}}{2 \cdot - 13}$

Step 8.1.3

Add $36$ and $156$ .

$u = \frac{6 \pm \sqrt{192}}{2 \cdot - 13}$

Step 8.1.4

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 8.1.4.1

Factor $64$ out of $192$ .

$u = \frac{6 \pm \sqrt{64 (3)}}{2 \cdot - 13}$

Step 8.1.4.2

Rewrite $64$ as $8^{2}$ .

$u = \frac{6 \pm \sqrt{8^{2} \cdot 3}}{2 \cdot - 13}$

Step 8.1.5

Pull terms out from under the radical.

$u = \frac{6 \pm 8 \sqrt{3}}{2 \cdot - 13}$

Step 8.2

Multiply $2$ by $- 13$ .

$u = \frac{6 \pm 8 \sqrt{3}}{- 26}$

Step 8.3

Simplify $\frac{6 \pm 8 \sqrt{3}}{- 26}$ .

$u = \frac{3 \pm 4 \sqrt{3}}{- 13}$

Step 8.4

Move the negative in front of the fraction.

$u = - \frac{3 \pm 4 \sqrt{3}}{13}$

Step 9

The final answer is the combination of both solutions.

$u = - \frac{3 + 4 \sqrt{3}}{13}, - \frac{3 - 4 \sqrt{3}}{13}$

Step 10

Substitute $\cos (x)$ for $u$ .

$\cos (x) = - \frac{3 + 4 \sqrt{3}}{13}, - \frac{3 - 4 \sqrt{3}}{13}$

Step 11

Set up each of the solutions to solve for $x$ .

$\cos (x) = - \frac{3 + 4 \sqrt{3}}{13}$

$\cos (x) = - \frac{3 - 4 \sqrt{3}}{13}$

Step 12

Solve for $x$ in $\cos (x) = - \frac{3 + 4 \sqrt{3}}{13}$ .

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Step 12.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- \frac{3 + 4 \sqrt{3}}{13})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arccos (- \frac{3 + 4 \sqrt{3}}{13})$ .

$x = 2.43983383$

Step 12.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 2.43983383$

Step 12.4

Solve for $x$ .

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Step 12.4.1

Remove parentheses.

$x = 2 (3.14159265) - 2.43983383$

Step 12.4.2

Simplify $2 (3.14159265) - 2.43983383$ .

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Step 12.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.43983383$

Step 12.4.2.2

Subtract $2.43983383$ from $6.2831853$ .

$x = 3.84335147$

Step 12.5

Find the period of $\cos (x)$ .

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Step 12.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 12.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 12.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2.43983383 + 2 π n, 3.84335147 + 2 π n$ , for any integer $n$

Step 13

Solve for $x$ in $\cos (x) = - \frac{3 - 4 \sqrt{3}}{13}$ .

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Step 13.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- \frac{3 - 4 \sqrt{3}}{13})$

Step 13.2

Simplify the right side.

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Step 13.2.1

Evaluate $\arccos (- \frac{3 - 4 \sqrt{3}}{13})$ .

$x = 1.26382862$

Step 13.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 1.26382862$

Step 13.4

Solve for $x$ .

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Step 13.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.26382862$

Step 13.4.2

Simplify $2 (3.14159265) - 1.26382862$ .

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Step 13.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.26382862$

Step 13.4.2.2

Subtract $1.26382862$ from $6.2831853$ .

$x = 5.01935668$

Step 13.5

Find the period of $\cos (x)$ .

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Step 13.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 13.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 13.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 1.26382862 + 2 π n, 5.01935668 + 2 π n$ , for any integer $n$

Step 14

List all of the solutions.

$x = 2.43983383 + 2 π n, 3.84335147 + 2 π n, 1.26382862 + 2 π n, 5.01935668 + 2 π n$ , for any integer $n$